"Éva is waiting for a doctor."
Translation:Éva vár egy orvost.
If the doctor was waiting for Eva, would her name (as a proper noun) alter the same way as any other noun, ie "Az orvos vár Évát?"
Yes, it would be "Évát". Proper nouns are treated very much the same as regular nouns, except in writing the suffix is sometimes hyphenated.
Other than that, Éva is waiting for "a" doctor, an indefinite object. The doctor, however, will probably wait for a specific Éva. So, correctly, there is no "egy". But the conjugation also changes (sorry), it will follow the 'definite conjugation', kind of like a more transitive form of the verb. The result is "várja".
"Az orvos Évát várja" or "Az orvos várja Évát".
Éva could also wait for the doctor. In that case:
"Éva várja az orvost".